For questions regarding harmonic functions.

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1answer
15 views

Are there identities which will make calculating $\Delta \frac{e^{ik|x|}}{|x|}$ more efficient and clearer?

Let $x = (x_1, x_2, x_3)$ be a point in $\mathbb{R}^3$. I am trying to calculate $$\Delta \frac{e^{ik|x|}}{|x|}$$ Doing this 'manually' be calculating second derivatives is really long and tedious ...
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2answers
27 views

Laplace transform

Hello how can I solve this problem $ty''-y'=t^2$, $y (0)=0 $ I though about several ways like $y''-y'/t =t$ but I did not know how to solve the integral of $( y'/t) $. Then I thought about ...
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0answers
17 views

Stuck while trying to show fundamental solution $\frac{1}{2\pi}\ln|x|$ satisfies $\Delta \phi(x) = \delta(x)$ in 2d?

I am trying to show that the fundamental solution to the Laplacian in 2D satisfies $$\Delta \phi(x) = \delta(x)$$ where $x = (x_1, x_2) \in \mathbb{R}^2$. So the fundamental solution in 2D is $\...
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1answer
36 views

Properties of harmonic function on $\mathbb{R}^2$

Assume $f$ is harmonic on $\mathbb{R}^2$. I want to prove that if there exists a constant $M$ such that $f(x,y) \geq M$ for all $(x,y)\in \mathbb{R}^2$, then $f$ must be a constant fuction. I'm ...
3
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1answer
52 views

Computing $\int_D x^3-3xy^2\,{\rm d}x\,{\rm d}y$ using the mean value property.

I am asked to compute $$\int_D x^3-3xy^2\,{\rm d}x\,{\rm d}y,$$where $D = \{ (x,y) \mid (x+1)^2+y^2 \leq 9, \text{and }(x-1)^2+y^2 \geq 1 \}$. Granted, $u(x,y) = x^3-3xy^2$ is harmonic (it is the real ...
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0answers
29 views

What does it mean for a complex-valued function to be bounded above (or below)?

I was reading about the maximum-minimum principle for harmonic functions in my lecture notes, and it was formulated like this: Let $\phi$ be harmonic in a simply-connected domain $D$. If $\phi$ is ...
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1answer
27 views

Relation between Poisson kernel and harmonic measure

If $D$ is a domain in the complex plane bounded by a Jordan curve $J$, what's the relation between the harmonic measure and the Poisson kernel on the boundary? More specifically, if $z_0 \in D$ and $...
3
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1answer
117 views

Harmonic function with injective boundary conditions is an immersion?

This question is in some sense a continuation of this question, though more focused in its scope. Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we ...
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1answer
45 views

Does harmonic decomposition preserve immersions?

In a nutshell: If we start from an immersion, and look at its harmonic decomposition, are the components also immersions? Details: Let $(M,g)$ be an $n$-dimensional, compact Riemannian manifold with ...
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1answer
31 views

The value of a harmonic function in the interior of a unit disk

Let $u(z)$ be a bounded harmonic function in $D$ such that the limit $$\lim_{r→1^-}u(re^{iφ})$$ is equal to 1 when for $0 < φ < π$ and to 0 for $π < φ < 2π$. Find $u(1/2)$. To ...
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1answer
61 views

Separation of variables for PDE: dividing by zero?

This feels like a question that is both simple and duplicate but I can't find an answer or a previous version of the question. Suppose we are given some PDE, for example Laplace's Equation in polar ...
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1answer
20 views

Use result that composition of analytic functions is harmonic to find harmonic conjugate of $e^{-2xy}\sin (x^{2}-y^{2})$

Using this result, I need to find a harmonic conjugate for $e^{-2xy}\sin(x^{2}-y^{2})$. In that result, I'm supposing that $s = e^{-2xy}$ and $t = \sin(x^{2}-y^{2})$, but I really don't know how to ...
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0answers
19 views

3 dimensional harmonic conjugates?

An $n$ dimensional harmonic function is defined to be a real valued function $f$ in $\mathbb{R}^n$ such that $\nabla^2 f = 0 $. Equivalently, $f$ is the scalar potential of a conservative vector field ...
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1answer
92 views

Show composition of harmonic function and analytic function is harmonic without calculating 2nd derivatives

This is not a duplicate. I realize similar questions to this have been asked, but what I am asking is slightly different: I need to prove the following, arguing by complex differentiability only, and ...
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0answers
16 views

Critical points of a harmonic function

Suppose $\phi$ is harmonic on some compact, connected region of $\mathbb{R}^3$. Is there an algorithm that is guaranteed to find all critical points of $\phi$? (Obviously, these will all be saddle ...
3
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1answer
39 views

Find a harmonic conjugate for the function $u(x,y)=x^{3}+Axy^{2}$ if one exists - $u(x,y)$ might not be harmonic?

For $u(x,y) = x^{3} + Axy^{2}$, where $A$ is a real number, I need to find a harmonic conjugate, or if one does not exist, show that it does not exist. I began my approach to this problem by first ...
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0answers
39 views

Square integrable functions on the unit ball

In one dimension, the space $L^{2}([0,1], dx)$ of complex-valued square integrable functions on $[0,1]$ is well known to be separable, and sine and cosine provide an explicit Hilbert basis for it. My ...
2
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1answer
33 views

Find a harmonic function which goes to $0$ on the boundary, which is not identically $0$

Find a function which is harmonic on the area bounded by positive x axis and the line $y=x$, which goes to $0$ on the boundary, which is not identically $0$. Why doesn't it violate the max/min ...
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1answer
43 views

Intuition behind $\nabla \cdot \frac{1}{\rho}\nabla$?

Oftentimes instead of the Laplacian I notice the very similar operator $$\nabla \cdot \frac{1}{\rho}\nabla$$ What is the intuition behind this operator? How does it differ intuitively from the ...
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2answers
65 views

Fourier Transform to solve Laplace's equation in cylindrical coordinates

I am trying to solve $\nabla^2 u = 0$ in cylindrical polar coordinates (and radial symmetry) $$ \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{\partial^2 u}{\...
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1answer
13 views

Laplace's equation 2 variable PDE/chain rule show function is a solution

The question is: 'Show that if f(x,y) is harmonic, then $f(x^2-y^2,2xy)$ is also harmonic using Laplace's equation: $\frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2} = 0 $. I end ...
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1answer
101 views

Find Green's function of quarter-plane with method of images

Consider a domain $$D= \{(x,y) : x>0, y>0 \}$$ Let $\textbf x = (x,y)$ and $\xi =(\xi_x , \xi_y)$. Find the Green's function, $G(\textbf x , \xi)$ such that $$\nabla ^2 G=\delta (\textbf x - \xi)...
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1answer
23 views

Inequality involving harmonic functions over the ball and half ball

Let $B\subset \mathbb R^2$ be a unit ball. Let $B^+:=B\cap \{x_2\geq 0\}$ where we set $x=(x_1,x_2)\in \mathbb R^2$. Let $\omega\in C^1(\partial B)$ be given such that $|\nabla \omega|>0$ for all $...
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0answers
30 views

Verify that Poisson's integral formula is harmonic on the unit disk

Let $D = \{ z : |z| < 1\}$ be the unit disk and let $z \in D$ and $e^{i\theta} \in \partial D$. Let $h(e^{i\theta})$ be a function defined on $\partial D$ and let $\tilde h(z)$ be defined in $D$. ...
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1answer
38 views

Greens function for 2d laplace equation with neumann boundary conditions

I have a domain, $ D : {(x,y) : x>0 , y>0}$ Let $ \mathbf{x}= (x,y) $ and $\mathbf{\xi}= (\xi_x, \xi_y)$, The Greens function satisfying: $$\nabla^2G = \delta(\mathbf{x} - \mathbf{\xi} )$$ ...
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0answers
14 views

Is it posible to solve the Laplce equation on an open set with dirac delta boundary conditions?

By dirac deta boundary conditions, i mean there is an $y \in \partial D$ such that the value on that point is $\infty$ and 0 elsewhere. Intuitivelly i would think it should be true: the Laplace ...
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2answers
51 views

Show that $z^n+c$ is harmonic, $c\in\mathbb C$.

I would like to know how to show $$f(z) = z^n+c$$ for $c\in \mathbb C$ is harmonic over $D =\{|z|\leq r\}$. I know that if I express $z = x+iy$, then I can have $f=u+iv$, where $u$ and $v$ will be ...
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1answer
45 views

Intersections of the level curves of two (conjugate) harmonic functions

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ with $f(x,y)=e^{-x}(x\sin y-y\cos y)$. 1 Let $g$ be one of the conjugate harmonics of $f$ on $\mathbb{R}^2$ and assume the level curves of $f$ and $g$ ...
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1answer
23 views

What is the harmonic conjugate of $u=4xy-3x+5y$?

What is the harmonic conjugate of $u=4xy-3x+5y$? I got $u'x=4-y=v'y$ then I integrated $v'y$ to get $v= 2y^2-3y+h(x)$. Then I did $-u'y=v'x$ so, $5= h'(x)$ then I integrated $5$ with respect to $x$ ...
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1answer
32 views

Why is $\int_{B(0,\epsilon)}|\Phi(y)|dy$ bounded?

Why is $\int_{B(0,\epsilon)}|\Phi(y)|dy$ bounded ? If $\Phi$ is the fundamental solution of Laplace equation defined by $\Phi(x)=\begin{cases}-\frac1{2\pi}\log|x|& \text{if n=2}\\\frac{1}{n(n-2)\...
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0answers
21 views

harmonic measure circle

I'm trying to compare the probability of a particle (performing Brownian Motion), starting a large distance away from a circle, passing through a specific section of that circle is approximately the ...
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1answer
48 views

Mean value of a subharmonic function, divided by the logarithm of radius, has a limit

I am pretty stuck on a homework problem on harmonic functions, or rather subharmonic functions (which for us are allowed to take the value $-\infty$). The statement is as follows: Supper $u$ is ...
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0answers
80 views

Deduce the definition of a harmonic function in the context of a Markov Chain

We know from PDE that a harmonic function $f$ satisfies the mean value property, namely, $f(x)$ = $\frac{1}{\vert{B_r(x)}\vert}\int_{B_r(x)}f(y)dy$ where $B_r(x)$ is the ball about $x$ with radius $r$....
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1answer
89 views

A harmonic function with sublinear growth at infinity is constant

Show that if $u$ is harmonic in $\mathbb R^n$ and $u=o(|x|)$, then $u$ is a constant.(Hint: use the solid version of the mean value property $u(x)=\frac{1}{\omega R^n} \int _{B_{R(x)}} u(y)dy$, and ...
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0answers
21 views

biharmonic complex analysis

Complex analysis texts typically discuss analytic functions whose real and imaginary components are harmonic and satisfy the Laplace equation, $\nabla^2 f = 0$. I am working with a complex function ...
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1answer
24 views

let $f(x)$ is real polynomial. Can we say that $f(\left| x \right|)$ is subharmonic?

A continuous function $\varphi :\mathbb{R} \to \mathbb{C}$ is subharmonic if and only if, for any closed disc in $U$ with centre $\lambda_0$ and radius $r$,$$\varphi ({\lambda _0}) \le \frac{1}{{2\pi }...
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1answer
30 views

Is the product of two subharmonic function necessarily subharmonic?

Defin: A continuous function $\varphi :\mathbb{R} \to \mathbb{C}$ is subharmonic if and only if, for any closed disc in $U$ with centre $\lambda_0$ and radius $r$,$$\varphi ({\lambda _0}) \le \frac{1}...
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0answers
18 views

Prove $\lim |u(x,y)|e^{-|x|}=0$ for harmonic function $u$

Let $\epsilon >0$ and $\Omega \subset \{(x,y): -\frac{\pi}{2}+\epsilon<y<\frac{\pi}{2}-\epsilon\}$. Let $u$ be harmonic function in $\Omega$ such that $u=0$ on $\partial \Omega$. Prove that ...
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0answers
43 views

Summation of series involving $\sinh$ of a square root

In a physics problem that I assigned in one of my classes recently, the following series arises: $$ S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + ...
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1answer
43 views

Square integrable harmonic function

Let $u$ be harmonic function on $\mathbb{R}^n$ such that $\int_{\mathbb{R}^n} u^2 dx1...dxn < +\infty$ (1). I've got to prove that it implies $u=0$. My idea was to prove that (1) implies that $u$ ...
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0answers
20 views

Co-efficient Problem For Univalent harmonic functions on Unit disk

The Clunie Sheil Small conjecture for the second co-efficient of a univalent harmonic function on the unit disk is as follows:- Suppose, $h(z)+\overline{g(z)}$ is a one-one harmonic function on the ...
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0answers
54 views

Generalized Coupon Collector's Problem [duplicate]

I just read about the coupon collector's problem where you're trying to find out how many coupons you need to collect on average before you get one complete set. This turns out to be $nH(n)$. I was ...
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2answers
35 views

Derive mean value property of harmonic function using this method

We can view harmonic function $u(x,y,z)$ as a wave function that does not depend on time $t$. Let $\overline{u}(r)$ be its average value on the circle with radius $r$. Assume we have proved that it ...
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1answer
31 views

Laplace equation for a lens-shaped volume - any approximate analytical solutions?

I need to solve Laplace equation for a lens (drop on a surface) with constant boundary condition on the top surface and zero on the bottom surface (this is the simplest case, not the only one I ...
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0answers
42 views

Is this a right calculation of Fourier transform?

I am trying to solve: Calculate Fourier transform (on $-\infty < x < \infty $) of: $$ f(x) = \begin{cases} 1, & -L<x<L \\ 0, & |x|\geq L \end{cases} $$ My ...
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0answers
45 views

Why does solving the Laplace equation on a graph using the method of relaxation get different results than a spectral method?

When I use the method of relaxation to solve Laplace's equation on a graph where the boundary conditions are fixing a set of nodes to either 0 or 1 then the solution ends up being entirely between 0 ...
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0answers
15 views

Surface Integral of a harmonic function and mean value property

I want to find a general expression of the following integral, where $h$ is a harmonic function (we're in $\mathbb{R}^2$): $\int_{|x-y|\leq a^2}\frac{h(y)}{\sqrt{a^2-|x-y|^2}}dy$ I think I can ...
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0answers
21 views

Harmonic function on 3-dimensional lamplighter group

Can anyone give an example of a non-constant bounded harmonic function on the Lamplighter group $\mathbb{Z}_2 \wr \ \mathbb{Z}^3$? This function should exist, because the random walk on this group has ...
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0answers
25 views

Fourier transform on forced linear harmonic oscillator

I'm having trouble solving the following equation: $\ddot x + \omega^2 x = a \sin \Omega t$, where $t >0$, $\omega \neq \Omega$, $x(0+) = 0 = \dot x(0+)$. It is asked to be done by Fourier ...
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0answers
33 views

Function $\phi (x,y) \mapsto \phi (u,v)$ [ie:…] under the transformation $f(z) = u(x,y) + v(x,y)$ where $f(z)$ is analytic & $d_z \,f(z)$ is not $0$

I am just having a bit of trouble understanding what I am being asked. The ie: in title says $\phi [x(u,v), y(u,v)]$ Part a) is asking to do the laplacian, $\nabla^2_{x,y} \phi (x,y)$ and to write ...